Article
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Functional Continuous Uncertainty Principle
Version 1
: Received: 24 July 2023 / Approved: 24 July 2023 / Online: 25 July 2023 (04:15:34 CEST)
A peer-reviewed article of this Preprint also exists.
Krishna, K.M. Functional Continuous Uncertainty Principle. SSRN Electronic Journal 2023, doi:10.2139/ssrn.4519299. Krishna, K.M. Functional Continuous Uncertainty Principle. SSRN Electronic Journal 2023, doi:10.2139/ssrn.4519299.
Abstract
Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces. Let $(\{f_\alpha\}_{\alpha\in \Omega}, \{\tau_\alpha\}_{\alpha\in \Omega})$ and $(\{g_\beta\}_{\beta\in \Delta}, \{\omega_\beta\}_{\beta\in \Delta})$ be continuous p-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align}\label{CUE} \mu(\operatorname{supp}(\theta_f x))^\frac{1}{p} \nu(\operatorname{supp}(\theta_g x))^\frac{1}{q} \geq \frac{1}{\displaystyle\sup_{\alpha \in \Omega, \beta \in \Delta}|f_\alpha(\omega_\beta)|}, \quad \nu(\operatorname{supp}(\theta_g x))^\frac{1}{p} \mu(\operatorname{supp}(\theta_f x))^\frac{1}{q}\geq \frac{1}{\displaystyle\sup_{\alpha \in \Omega , \beta \in \Delta}|g_\beta(\tau_\alpha)|}. \end{align} where \begin{align*} &\theta_f: \mathcal{X} \ni x \mapsto \theta_fx \in \mathcal{L}^p(\Omega, \mu); \quad \theta_fx: \Omega \ni \alpha \mapsto (\theta_fx) (\alpha):= f_\alpha (x) \in \mathbb{K},\\ &\theta_g: \mathcal{X} \ni x \mapsto \theta_gx \in \mathcal{L}^p(\Delta, \nu); \quad \theta_gx: \Delta \ni \beta \mapsto (\theta_gx) (\beta):= g_\beta (x) \in \mathbb{K} \end{align*} and $q$ is the conjugate index of $p$. We call Inequality (\ref{CUE}) as \textbf{Functional Continuous Uncertainty Principle}. It improves the Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle obtained by M. Krishna in \textit{[arXiv:2304.03324v1, 2023]}. It also answers a question asked by Prof. Philip B. Stark to the author.
Keywords
Uncertainty Principle; Parseval Frame; Banach space
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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